124 The International Arab Journal of Information Technology, Vol. 6, No. 2, April 2009

Discrete Time NHPP Models for Software

Reliability Growth Phenomenon

Omar Shatnawi

Prince Hussein bin Abdullah Information Technology College, al-Bayt University, Jordan

Abstract: Nonhomogeneous poisson process based software reliability growth models are generally classified into two

groups. The first group contains models, which use the machine execution time or calendar time as a unit of fault

detection/removal period. Such models are called continuous time models. The second group contains models, which use the

number of test occasions/cases as a unit of fault detection period. Such models are called discrete time models, since the unit

of software fault detection period is countable. A large number of models have been developed in the first group while there

are fewer in the second group. Discrete time models in software reliability are important and a little effort has been made in

this direction. In this paper, we develop two discrete time SRGMs using probability generating function for the software

failure occurrence / fault detection phenomenon based on a NHPP namely, basic and extended models. The basic model

exploits the fault detection/removal rate during the initial and final test cases. Whereas, the extended model incorporates fault

generation and imperfect debugging with learning. Actual software reliability data have been used to demonstrate the

proposed models. The results are fairly encouraging in terms of goodness-of-fit and predictive validity criteria due to

applicability and flexibility of the proposed models as they can capture a wide class of reliability growth curves ranging from

purely exponential to highly S-shaped.

Keywords: Software engineering, software testing, software reliability, software reliability growth model, nonhomogeneous

poisson process, test occasions.

Received August 19, 2007; accepted December 9, 2007

1. Introduction

Software reliability assessment is important to evaluate

and predict the reliability and performance of software

system. The models applicable to the assessment of

software reliability are called Software Reliability

Growth Models (SRGMs). An SRGM provides a

mathematical relationship between time span of testing

or using the software and the cumulative number of

faults detected. It is used to assess the reliability of the

software during testing and operational phases.

An important class of SRGM that has been widely

studied is NonHomogeneous Poisson Process (NHPP).

It forms one of the main classes of the existing SRGM;

due to it is mathematical tractability and wide

applicability. NHPP models are useful in describing

failure processes, providing trends such as reliability

growth and fault-content. SRGM consider the

debugging process as a counting process characterized

by the mean value function of a NHPP. Software

reliability can be estimated once the mean value

function is determined. Model parameters are usually

determined using either Maximum Likelihood

Estimate (MLE) or least-square estimation methods [6,

13, 18].

NHPP based SRGM are generally classified into

two groups [2, 6, 17]. The first group contains models,

which use the execution time (i.e., CPU time) or

calendar time. Such models are called continuous time

models. The second group contains models, which use

the number of test cases as a unit of fault detection

period. Such models are called discrete time models,

since the unit of software fault detection period is

countable. A test case can be a single computer test run

executed in an hour, day, week or even month.

Therefore, it includes the computer test run and length

of time spent to visually inspect the software source

code [2, 6]. A large number of models have been

developed in the first group while fewer are there in

the second group due to the difficulties in terms of

mathematical complexity involved.

The utility of discrete time models cannot be

underestimated since the number of test cases (i.e.,

executed test run) is more appropriate measure of the

fault detection/removal period than the CPU/calendar

time used by continuous time model. These models

generally provide a better fit than their continuous time

counterparts because they are more realistic. Therefore,

in spite of difficulties in terms of mathematical

complexity involved, discrete time models are

proposed regularly.

Yamada and Osaki [17] discussed two classes of

models. One class describes the fault detection process

in which the expected number of faults per test case is

geometrically decreasing whereas, in the second is

proportional to the current fault-content and this class

Discrete Time NHPP Models for Software Reliability Growth Phenomenon 125

is known as exponential model. Kapur et al. [6]

proposed a model in which the testing phase is

assumed to have two different processes namely, fault

isolation and fault detection processes, this model is

known as delayed S-shaped model. Further they

proposed another model based on the assumption that

software may contain several types of faults. In these

models the fault removal process (i.e., debugging

process) is assumed to be perfect. But due to the

complexity of the software system and the incomplete

understanding of software requirements, specifications

and structure, the testing team may not be able to

remove the fault perfectly and the original fault may

remain or replaced by another fault. While the first

phenomenon is known as imperfect debugging. The

second is called error/fault generation.

The concept of imperfect debugging was first

introduced by Goel [3]. He introduced the probability

of imperfect debugging in Jelinski and Moranda model

[5]. Kapur et al. [6] introduced the imperfect

debugging in Goel and Okumoto model [4]. They

assumed that the fault removal rate per remaining

faults is reduced due to imperfect debugging. Thus the

number of failures observed by time infinity is more

than the initial fault content. Although these two

models describe the imperfect debugging phenomenon

yet the software reliability growth curve of these

models is always exponential. Moreover, they assume

that the probability of imperfect debugging is

independent of the testing time. Thus, they ignore the

role of the learning process during the testing phase by

not accounting for the experience gained with the

progress of software testing. Actually, the probability

of imperfect debugging is supposed to be a maximum

in the early stage of the testing phase and is supposed

to reduce with the progress of testing [1, 6, 7, 8, 10, 12,

13, 14, 15, 16].

In this paper, we propose two discrete time NHPP

based SRGMs for the situation given above. The

assumptions in this case are with respect to number of

test cases instead of time. The rest of this paper is

organized as follows. Section 2 derives the proposed

two flexible discrete time models. Sections 3 and 4

discuss the methods used for parameter estimation and

the criteria used for validation and evaluation of the

proposed models. The applications of the proposed

discrete to actual software reliability data through data

analyses and model comparisons are shown in section

5. We conclude this paper in section 6.

2. Software Reliability Modelling

2.1. Basic Discrete Time Model

2.1.1. Model Development

The main assumption of SRGM is that the failure

observation / fault detection depends linearly upon the

number of remaining faults [4]. This assumption

implies that all faults are equally likely to be detected.

In practical situation it has been observed that large

number of simple faults is easily detected at the early

stages of the testing, while the fault detection may

become extremely difficult in the later stages of the

testing phase. In this case, the fault detection rate has a

high value at the beginning of the testing as compared

to the value at the end of the testing. On the contrary, it

may happen that the detection of faults increases the

skill of the testing team and thus leading to increase in

efficiency. In other words, the testing team detects

more faults in less time. This implies that the value of

the fault detection rate at the end of the testing phase is

higher than it is value at the beginning of the testing.

Bittanti et al. [1] exploited this change in fault

detection rate, which they termed as the Fault

Exposure Coefficient (FEC) for their SRGM.

Through real data experiments and analysis on

several software development projects, the fault

detection rate has three possible trends as time

progresses; increasing, decreasing or constant [1, 10].

It decreases when the software has been used and

tested repeatedly, showing reliability growth. It can

also increase if the testing techniques/requirements are

changed, or new faults are introduced due to new

software features or imperfect debugging. Thus, we

treat the fault detection rate as a function of the number

of test cases to interpret these trends.

2.1.2. Model Assumptions, Notations, and

Formulation

• Failure observation/fault detection phenomenon is

modelled by NHPP.

• Software is subject to failures during execution

caused by faults remaining in the software.

• Each time a failure occurs, the fault that caused it is

immediately and perfectly detected, and no new

faults are introduced, i.e., the debugging process is

perfect.

a Initial fault content of the software.

b(n+1) Fault detection rate function is dependent on

the number of test cases and behaves linearly

on the number of faults remaining.

bi Initial fault detection rate.

bf Final fault detection rate.

m(n) The expected mean number of faults detected

by the nth test case.

Under the given model assumptions, the expected

cumulative number of faults detected between the nth

and (n+1)th test cases is proportional to the number of

faults remaining after the execution of the nth test run,

satisfies the following difference equation:

( ))n(ma)1n(b)n(m)1n(m −+=−+ (1)

126 The International Arab Journal of Information Technology, Vol. 6, No. 2, April 2009

The fault detection rate is given as a function of the

number faults detected, as in the following equation,

a

nmbbbnb ifi

)1()()1(

+−+=+ (2)

According to the values of bi and bf, we can

distinguish between the following cases [1]:

• Constant fault detection rate; bi=bf =b

• Increasing fault detection rate; bf >bi

• Decreasing fault detection rate; bi>bf

• Vanishing fault detection rate; bf =0, bi>0

By substituting equation 2 in the difference equation 1

and then solving it using Probability Generating

Function (PGF) with initial condition m(n=0)=0, one

can get the closed form solution as given below.

( )

n

f

i

if

n

f

bb

bb

banm

)1(1

)1(1)(

−−

+

−−= (3)

The structure of the model is flexible. The shape of

the growth curve is determined by the parameters bi

and bf and can be both exponential and S-shaped for

the four cases discussed above. In case of constant

fault detection rate, equation 3 can be written as

( )nbanm )1(1)( −−= (4)

If bf<bi, it is apparent from equation 1 that b(n+1)

decreases to zero more rapidly than linearly. The

smaller ratio bf/bi, the larger rate convergence of

equation 1. If bf=0, then equation 1 becomes:

( )( ))()1()()1( nmanmaa

bnmnm i −+−=−+ (5)

The solution of which is given by:

nb

nabnm

i

i

+=

1)( (6)

If bf>bi, then equation 3 has an inflection point and

the growth curve is S-shaped. Such behaviour of the

SRGM can be attributed to the increased skill of the

test team or a modification of the testing strategy.

The proposed discrete time model defined in

equation 3 is very interesting from various points of

view. Besides the above-mentioned interpretation as a

flexible S-shaped fault detection model, this model has

the exponential model [17] as special cases as given in

equation 4.

Note that this proposed basic discrete time model is

able to model both cases of strictly decreasing failure

intensity and the case of increasing-and-decreasing

failure intensity. None of the exponential model [17]

and the ordinary delayed S-shaped model [6] can do

both.

2.2. Extended Discrete Time Model

2.2.1. Model Development

During debugging process faults are identified and

removed upon failures. In reality this may not be

always true. The corrections may themselves introduce

new faults or they may inadvertently create conditions,

not previously experienced, that enable other faults to

cause failures. This results in situations where the

actual fault removals are less than the removal

attempts. Therefore, the fault detection rate is reduced

by the probability of imperfect debugging [6, 8, 13,

14]. Besides there is a good chance that some new

faults might be introduced during the course of

debugging.

The learning-process of the software developers has

also been studied [1, 6, 12, 16]. Leaning occurs if

testing appears to improve dynamically in efficiency as

one progress through the testing phase. Learning

usually manifests itself as a changing fault detection

rate. By assuming the fault detection rate is dependent

on the number of test cases, the role of the learning

process during the testing phase can be established.

The extended model proposed below incorporates

these three factors.

To avoid mathematical complexity many

simplifying assumptions have been taken while

developing the basic model. One of which is that the

number of initial fault-content is constant.

Modifications to the basic model have been proposed

for increasing over-all fault content, where a of

equation 1 is substituted by a(n) as given in equation 8.

The nature of a(n) depends upon various factors like

the skill of test team or testing strategy, number of test

cases, software size, and complexity. Hence no single

functional form can describe the growth in number of

faults during testing phase and debugging process. This

necessitates a modelling approach that can be modified

without unnecessarily increasing the complexity of the

resultant model [8].

Here, we show how this can be achieved for the

proposed basic discrete time model by changing the

fault detection rate. Before the spread of flexible

models, a number of models were proposed that linked

the fault detection rate to the initial fault content. But

basic model due to it is inherent flexibility could

describe many of them.

Hence it is imperative to capture this flexibility and

in this paper we do so by proposing a logistic number

of test cases dependent for fault detection rate as

proposed in equation 9.

2.2.2. Model Assumptions

In addition to basic discrete time model assumptions

except for assumption 3, we have:

• Over-all fault content is linearly dependent on the

number of test cases.

Discrete Time NHPP Models for Software Reliability Growth Phenomenon 127

• Fault detection rate is number of test cases

dependent function with an inflection S-shaped

model.

• Fault introduction rate is a linear function of the

number of test cases.

• Faults can be introduced during the testing phase

and the debugging process may not lead to the

complete removal of the faults, i.e., the debugging

process is imperfect.

2.2.3. Model Notations

In addition to the basic discrete time model notations

we include the following:

a(n) Over-all fault-content dependent on the

number of test cases, which includes initial

fault-content and the number of faults

introduced.

b(n+1) Fault detection rate dependent on the number of

test cases.

α Fault introduction rate per detected faults per

test case.

p The probability of fault removal on a failure,

i.e., the probability of perfect debugging

Fault detection rate function is dependent on

the number of test cases and behaves linearly

on the number of faults remaining.

2.2.4. Model Formulation

Under the above extended model assumptions, the

expected cumulative number of faults detected

between the nth and (n+1)

th test cases is proportional to

the number of faults remaining after the execution nth

test run, satisfies the following difference equation:

( ))()()1()()1( nmnanbnmnm −+=−+ (7)

Both a(n) and b(n+1) are number of test cases

dependent functions. An increasing a(n) implies an

increasing total number of faults, and thus reflects fault

generation. Whereas, b(n+1) is an S-shaped curve that

can capture the learning process of the software testers,

and this function is affected by the probability of fault

removal on a failure and they are giving as:

)1()( nana α+= (8)

and

1)1(1

)1(+−

−+

=+n

f

i

if

f

pbb

bb

pbnb (9)

By substituting equations 9 and 8 in 7 and then

solving it using PGF with initial condition m(n=0)=0,

after tedious algebraic manipulations, one can get the

closed form solution as given below.

( )[

+

−−−

×

−−

+

=

npb

pbpb

pbb

bb

anm

f

fn

f

n

f

i

if

αα

)1(1

)1(1

)( (10)

Fault generation and imperfect debugging with leaning

are integrated to form the extended discrete time model

as given in equation 10.

According to the values of parameters, we can

distinguish between the following cases:

• Constant fault detection rate (bi=bf=b), no faults

introduced (α=0), and prefect debugging (p=1)

• No faults introduced (α=0), and prefect debugging

(p=1).

In case 1, equation 10 can reduce to equation 4.

Whereas in case 2, it reduces to equation 3.

3. Parameter Estimation

MLE method is used to estimate the unknown

parameters of the developed framework. Since all data

sets used are given in the form of pairs

(ni,xi)(i=1,2,…,f), where xi is the cumulative number of

faults detected by ni test cases (0<n1<n2<…<nf) and ni

is the accumulated number of test run executed to

detect xi faults.

The likelihood function L for the unknown

parameters with the superposed mean value function is

given as

( ) [ ]

( )( ))()(exp

)!(

)()(),(|

1

1 1

11

−

= −

−−

−−

×−

−=∏

−

ii

f

i ii

xx

iiii

nmnm

xx

nmnmxnparmatersL

ii

(11)

Taking natural logarithm of equation 11 we get

[ ]

{ } [ ]∑

∑

=−−

−=

−

−−−

−−−=

f

i

iiii

ii

f

i

ii

xxnmnm

nmnmxxL

1

11

1

1

1

)!(ln)()(

)()(ln)(ln (12)

The MLE of the SRGM parameters can be obtained

to by maximizing L in equation 11 or 12 with respect

to the following constraints: (a,bi,bf>0, 0<p≤1, α≥0).

4. Model Validation and Comparison

Criteria

4.1. Model Validation

To check the validity of the proposed discrete time

models to describe the software reliability growth, they

have been tested on four Data Sets (DS) cited from real

software development projects.

128 The International Arab Journal of Information Technology, Vol. 6, No. 2, April 2009

The first DS-I was collected from test of a network-

management system at AT&T Bell Laboratories, after

it was tested for 20 weeks in which 100 faults were

detected [13]. The second DS-II was collected during

21 days of testing, 46 faults were detected [12]. The

third DS-III was for a radar system of size 124 KLOC

after it was tested for 35 months in which 1301 faults

were detected [2]. The fourth DS-IV had been

collected during 25 days of testing in which 136 faults

were detected [13].

4.2. Comparison Criteria

The performance of an SRGM judged by its ability to

fit the past software reliability data (goodness-of-fit)

and to predict satisfactorily the future behavior from

present and past data (predictive validity) [6, 11].

4.2.1. Goodness of Fit

• The Sum of Squared Error (SSE). The difference

between the simulated data m^(ni) and the observed

(reported data) xi is measured by the SSE as,

( )∑=

−=f

i

ii xnmSSE1

2)(ˆ (13)

where f is the number of observations. The lower value

of SSE indicates less fitting error, thus better

goodness-of-fit.

• The Akaike Information Criterion (AIC). This

criterion was first proposed as SRGM model

selection tool by [9]. It is defined as:

AIC= -2(value of max. log likelihood function) +

2(number of parameters used in the model) (14)

Lower value of AIC indicates more confidence in the

model thus a better fit and predictive validity.

• Coefficient of multiple determination (R2). This

measure can be used to investigate whether a

significant trend exists in the observed failure

intensity. This coefficient is defined as the ratio of

the Sum of Squares (SS) resulting from the trend

model to that from a constant model subtracted from

1, that is:

SScorrected

SSresidualR −=12 (15)

R2 measures the percentage of the total variation about

the mean accounted for by the fitted curve. It ranges in

value from 0 to 1. Small values indicate that the model

does not fit the data well [11].

4.2.2. Predictive Validity

Predictive validity is defined as the ability of the model

to determine the future failure behaviour from present

and past failure behaviour [11]. The relative prediction

fault (RPF) is defined as,

f

ff

x

xnmRPF

−=

)(ˆ (16)

where xf is the cumulative number of faults removed

after the execution of the last test run nf and m^(nf), is

the estimated value of the SRGM m(nf), which

determined using the actually observed data up to an

arbitrary test case ne(≤nf).

If the RPF value is negative/positive the model is

said to underestimate/ overestimate the fault removal

process. A value close to zero indicates more accurate

prediction, thus more confidence in the model. The

value is said to be acceptable if it is within (±10%) [6].

5. Data Analysis and Model Comparison

5.1. For Basic Discrete Time Model

5.1.1. Goodness of Fit Analysis

Using MLE method, the estimated values of the

proposed basic discrete time model parameters for DS-

I and DS-II are given in Table 1. According to the

estimated values of the initial and final fault detection

rates (bi and bf), the skill of the test-team does improve

with time in both DS-I and DS-II. That is why the fault

detection/removal process resembles an S-shaped

growth curve in both DS-I and DS-II.

Table 1. Parameters estimations for DS-I and DS-II.

Parameter Estimation

Model Data

Set a bi bf

DS-I 111 .0717 .1581 Proposed

Basic DS-

II 59 .0167 .1550

The fitting of the proposed model to both DS-I and

DS-II are graphically illustrated in Figures 1 and 2

respectively. It is clearly seen from both the figures

that the proposed model fits both DS-I and DS-II

excellently. This highlights it is flexibility.

0

28

56

84

112

0 5 10 15 20

Test Cases (w eeks)

Cum

ulativ

e F

aults

Actual Data Estimated Values

Figure 1. Goodness of fit (DS-I).

Discrete Time NHPP Models for Software Reliability Growth Phenomenon 129

0

12

24

36

48

0 7 14 21

Test Cases (months)

Cumulative Faults

Actual Data Estimated Values

Figure 2. Goodness of fit (DS-II).

Comparison of the proposed model and well-

documented discrete time NHPP based SRGMs in

terms of goodness of fit is given in Tables 2 and 3 for

DS-I and DS-II respectively. Note that during the

estimation process of models under comparison it is

observed that the exponential model [17] fails to give

any plausible result as it over estimates the fault-

content and no estimates were obtained for DS-II. It is

clearly seen from both the Tables 2 and 3 that the

proposed model is the best among the models under

comparison in terms of SSE, AIC, and R2 metrics

values, which is very encouraging.

Table 2. Goodness of fit for DS-I.

Parameter

Estimation

Comparison

Criteria Models under

Comparison a b SSE AIC R2

Exponential [17] 130 .0798 232 92 .9857

Delayed S-shaped [6] 106 .2165 357 99 .9781

Proposed Basic See Table 1. 180 90 .9890

Table 3. Goodness of fit for DS-II.

Parameter

Estimation

Comparison

Criteria Models under

Comparison a b SSE AIC R2

Exponential [17] * * * * *

Delayed S-shaped [6] 84 .0831 28 79 .9938

Proposed Basic See Table 1. 25 77 .9944

Hence, the proposed basic discrete time model fits

better than existing models on both DS-I and DS-II.

5.1.2. Predictive Validity Analysis

DS-I and DS-II are truncated into different proportions

and used to estimate the parameters of the proposed

basic discrete time model. For each truncation, one

value of RPE ratio is obtained.

Figures 3 and 4 graphically illustrate the results of

the predictive validity. It is observed that the predictive

validity of the proposed model varies from one

truncation to another. The RPE ratio of the proposed

model overestimates the fault removal process in DS-I

and DS-II except when the testing progress ratio is

about 55% and 50% it underestimates the process in

DS-II.

-0.2

-0.1

0

0.1

0.2

50% 60% 70% 80% 90% 100%

Testing Progress Ratio

RPE

Predictive Validity Curve

Figure 3. Predictive validity (DS-I).

-0.2

-0.1

0

0.1

0.2

50% 60% 70% 80% 90% 100%

Testing Progress Ratio

RPE

Predictive Validity Curve

Figure 4. Predictive validity (DS-II).

It is clearly seen from both Figures 3 and 4 that 55%

of the total test time is sufficient to predict the future

behaviour of the fault removal process reasonably for

DS-I and DS-II.

5.2. For Extended Discrete Time Model

5.2.1. Goodness of Fit Analysis

Using MLE method, the estimated values of the

proposed extended discrete time model parameters for

DS-III and DS-IV are given in Table 4. According to

the estimated values of the initial and final fault

detection rates (bi and bf), the skill of the test-team

does improve with time in DS-III and in DS-IV does

not. That is why the fault detection process resembles

an S-shaped growth curve in DS-III and an exponential

curve in DS-IV. According to the estimated values of

the fault introduction rate parameter (α) the fault

detection process (i.e., the debugging process) in DS-

III is perfect and no fault introduced during debugging,

whereas in DS-IV was not.

Table 4. Parameters estimations for DS-III & -IV.

Parameter Estimation Model

Data

Set a bi bf p α

DS-III 1352 .0087 .1832 .9922 0 Proposed

Extended DS-IV 156 .1525 .0005 .9965 .0036

The fitting of the proposed model to both DS-III and

DS-IV are graphically illustrated in Figures 5 and 6. It

may be noticed that the relationship between the

cumulative number of faults and the number of test

cases vary from purely exponential to highly S-shaped.

It is clearly seen from both the figures that the

130 The International Arab Journal of Information Technology, Vol. 6, No. 2, April 2009

proposed model fits both the DS-III and DS-IV

excellently. This highlights it is flexibility.

0

328

656

984

1312

0 7 14 21 28 35

Test Cases (months)

Cumulative Faults

Actual Data Estimated Values

Figure 5. Goodness of fit for DS-III.

0

34

68

102

136

0 5 10 15 20 25

Test Cases (days)

Cumulative Faults

Actual Data Estimated Values

Figure 6. Goodness of fit (DS-IV).

Comparison of the proposed model and well-

documented discrete time SRGM based on NHPP in

terms of goodness of fit is given in Tables 5 and 6 for

DS-III and DS-IV respectively. Note that during the

estimation process of models under comparison it is

observed that the exponential model [17] fails to give

any plausible result as it over estimates the fault-

content (a) and no estimates were obtained for DS-III.

It is clearly seen from both the Tables 5 and 6 that the

proposed model is the best among the models under

comparison in terms of SSE, AIC, and R2 metrics

values, which is very encouraging. Hence, the

proposed extended discrete time model fits better than

existing models on both DS-III and DS-IV.

Table 5. Goodness of fit for DS-III.

Parameter

Estimation

Comparison

Criteria Models under

Comparison

a b SSE AIC R2

Exponential [17] * * * * *

Delayed S-shaped [6] 1735 .0814 107324 518 .9856

Proposed Extended See Table 5. 7133 342 .9990

Table 6. Goodness of fit for DS-III.

Parameter

Estimation

Comparison

Criteria Models under

Comparison

a b SSE AIC R2

Exponential [17] 136 .1291 766 119 .9664

Delayed S-shaped [6] 126 .2763 2426 176 .8936

Proposed Extended See Table 5. 306 116 .9866

5.2.2. Predictive Validity Analysis

DS-III and DS-IV are truncated into different

proportions and used to estimate the parameters of the

proposed extended discrete time model. For each

truncation, one value of RPE ratio is obtained.

Figures 7 and 8 graphically illustrate the results of

the predictive validity. It is observed that the predictive

validity of the proposed model varies from one

truncation to another. The RPE ratio of the proposed

model overestimates the fault removal process in DS-

III and DS-IV except when the testing progress ratio is

about 65% and 60% it underestimates the process in

DS-IV.

-0.2

-0.1

0

0.1

0.2

50% 60% 70% 80% 90% 100%

Testing Progress Ratio

RPE

Predictive Validity Curve

Figure 7. Predictive validity (DS-III).

-0.2

-0.1

0

0.1

0.2

50% 60% 70% 80% 90% 100%

Testing Progress Ratio

RPE

Predictive Validity Curve

Figure 8. Predictive validity (DS-IV).

It is clearly seen from both Figures 7 and 8 that 50%

of the total test time is sufficient to predict the future

behaviour of the fault removal process reasonably for

DS-III and DS-IV.

6. Conclusion

In this paper, newly developed discrete time SRGM

based on NHPP to describe a variety of reliability

growth and the increased skill (efficiency) of the

testing team or a modification of the testing strategy

during testing phase, are proposed.

The proposed discrete time models have been

validated and evaluated on actual software reliability

data cited from real software development projects and

compared with existing discrete time NHPP based

models. The results are encouraging in terms of

goodness of fit and predictive validity due to their

applicability and flexibility. Hence, we conclude that

Discrete Time NHPP Models for Software Reliability Growth Phenomenon 131

the two proposed discrete time models not only fit the

past well but also predict the future reasonably well.

Acknowledgements

I take this opportunity to thank Prof. P. K. Kapur, Dr.

A. K Bardhan, and Dr. P. C. Jha of Delhi University

for their support every moment I sought. The

suggestion, comments, and criticisms of these people

have greatly improved this manuscript.

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Omar Shatnawi received his PhD,

in computer science and his MSc in

operational research from

University of delhi in 2004 and

1999, respectively. Currently, he is

head of Department of Information

Systems at al-Bayt University. His

research interests are in software engineering, with an

emphasis on improving software reliability and

dependability.